March 20, 2019
We start with a causal claim
B/C of FPCI: imperfectly test hypotheses using correlation of observed values of \(X\) and \(Y\)
Equivalently:
Adjustment-based approaches start from the…
Sometimes called the "method of difference" (via John Stuart Mill), this assesses whether \(X\) causes \(Y\) by…
What causes the spread of cholera?
Contaminated water causes cholera outbreaks
19th Century London saw repeated outbreaks of cholera, with mass death
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John Snow, MD
Snow could compare places with contaminated water with those that did not, but there is a possibility of confounding:
Comparing observed water quality and cholera in:
would not rule out miasma as explanation for cholera.
Near the outbreak
| Brewers | Broad St. Residents | |
|---|---|---|
| Water Source | Brewery Well/ Beer (Clean) |
Pump (Contam.) |
| Location | Near pump | Near pump |
| Timing | Aug. 1854 | Aug. 1854 |
| Miasmas? | Yes? | Yes? |
| Cholera | No | Yes |
Far from outbreak
| Lady and Niece | Non-Soho Residents | |
|---|---|---|
| Water Source | Broad Street Pump (Contam.) |
Another Pump (Clean) |
| Location | Far from Broad St. | Far from Broad St. |
| Timing | Aug. 1854 | Aug. 1854 |
| Miasmas? | No | No |
| Cholera | Yes | No |
A more general approach to the comparative method is:
when we observe \(X\) and \(Y\) for multiple cases, we examine the correlation of \(X\) and \(Y\) within groups of cases that have the same values of confounding variables \(W, Z, \ldots\).
How does conditioning solve the problem? Equivalently:
Sometimes we think about "conditioning" like this:
But all else does not need to be equal:
Does having "sanctuary" policy increase crime?
| \(i\) | \(Sanctuary_i\) | \(Crime_i^{Sanct.}\) | \(Crime_i^{No \ Sanct.}\) | \(S_i - No \ S_i\) |
|---|---|---|---|---|
| 1 | \(\mathbf{Yes}\) | \(\mathbf{35}\) | 40 | -5 |
| 2 | \(\mathbf{Yes}\) | \(\mathbf{35}\) | 40 | -5 |
| 3 | \(\mathbf{Yes}\) | \(\mathbf{5}\) | 10 | -5 |
| 4 | \(\mathbf{No}\) | 35 | \(\mathbf{40}\) | -5 |
| 5 | \(\mathbf{No}\) | 5 | \(\mathbf{10}\) | -5 |
| 6 | \(\mathbf{No}\) | 5 | \(\mathbf{10}\) | -5 |
If we simple look at the unadjusted difference in crime in sanctuary vs. non-sanctuary counties, we would get:
\[\frac{\overbrace{35 + 35 + 5}^{\text{Mean Crime in Sanctuary}}}{3} - \frac{\overbrace{40 + 10 + 10}^{\text{Mean Crime w/out Sanctuary}}}{3} = \frac{15}{3} \neq -5\]
This is bias and must result from confounding
\(Urban_i\) is a confounding variable. Why?
| \(i\) | \(Sanctuary_i\) | \(Urban_i\) | \(Crime_i^{Sanct.}\) | \(Crime_i^{No \ Sanct.}\) | \(S_i - No \ S_i\) |
|---|---|---|---|---|---|
| 1 | \(\mathbf{Yes}\) | \(\mathbf{Yes}\) | \(\mathbf{35}\) | 40 | -5 |
| 2 | \(\mathbf{Yes}\) | \(\mathbf{Yes}\) | \(\mathbf{35}\) | 40 | -5 |
| 3 | \(\mathbf{Yes}\) | \(\mathbf{No}\) | \(\mathbf{5}\) | 10 | -5 |
| 4 | \(\mathbf{No}\) | \(\mathbf{Yes}\) | 35 | \(\mathbf{40}\) | -5 |
| 5 | \(\mathbf{No}\) | \(\mathbf{No}\) | 5 | \(\mathbf{10}\) | -5 |
| 6 | \(\mathbf{No}\) | \(\mathbf{No}\) | 5 | \(\mathbf{10}\) | -5 |
If we condition on \(Urban\) (compare \(X\) and \(Y\) for cases with same value of \(Urban\)), we can remove the bias.
| \(i\) | \(Sanctuary_i\) | \(Urban_i\) | \(Crime_i^{Sanct.}\) | \(Crime_i^{No \ Sanct.}\) | \(S_i - No \ S_i\) |
|---|---|---|---|---|---|
| 1 | \(\mathbf{Yes}\) | \(\mathbf{Yes}\) | \(\mathbf{35}\) | 40 | -5 |
| 2 | \(\mathbf{Yes}\) | \(\mathbf{Yes}\) | \(\mathbf{35}\) | 40 | -5 |
| 4 | \(\mathbf{No}\) | \(\mathbf{Yes}\) | 35 | \(\mathbf{40}\) | -5 |
| 3 | \(\mathbf{Yes}\) | \(\mathbf{No}\) | \(\mathbf{5}\) | 10 | -5 |
| 5 | \(\mathbf{No}\) | \(\mathbf{No}\) | 5 | \(\mathbf{10}\) | -5 |
| 6 | \(\mathbf{No}\) | \(\mathbf{No}\) | 5 | \(\mathbf{10}\) | -5 |
If we condition on \(Urban\), we get the true effect of \(-5\):
\[\frac{\overbrace{35 + 35}^{\text{Sanctuary (Urban)}}}{2} - \frac{\overbrace{40}^{\text{w/out Sanctuary (Urban)}}}{1} = -5\]
\[\frac{\overbrace{5}^{\text{Sanctuary (Not Urban)}}}{1} - \frac{\overbrace{10 + 10}^{\text{w/out Sanctuary (Not Urban)}}}{2} = -5\]
When we condition on \(W\) (Urban), by looking at cases where \(W\) is the same:
Without these two relationships, confounding does not happen, there is no bias.
in order for adjustment solutions to confounding/bias to work?